The false dichotomy between knowledge and creativity

There is currently a what I consider inane discussion going on in Sweden about our at best mediocre PISA (Program for International Student Assessment) scores. The arguments presented about why the scores do not matter are so poor that even the most ardent opponent of standardised testing has to recoil in horror. I’m personally not a huge fan of standardised tests, and I completely agree that some things cannot be measured accurately with standardised tests. We do not want the school system to evolve into the American one with full of multiple choice tests and little room for other types of evaluation. However, tests like PISA do give a very accurate measure of things that are simple to test, like reading comprehension and simple arithmetic skills. A strong decline in absolute reading and math scores over a comparatively short period of time (12 years) is a cause for concern. Because it is not just the relative ranking that has dropped: the absolute scores have dropped as well. For example: reading comprehension scores decreased from an average of 516 points in 2000 to an average of 483 points in 2012. And the decline is steady.

Every 6 months or so, like clockwork, someone comes out to say that we shouldn’t actually want our students to do well on the PISA. The reason why is that PISA measures knowledge, and we want our students to be creative, not just able to regurgitate knowledge. For those of you who can read Swedish, the latest example of this argument is here. The argument is the same as ever: creativity matters more than knowledge, teaching kids knowledge stifles creativity, therefore we should not care that a large proportion of kids cannot read anything more complex than a Donald Duck comic. There is also some fear mongering about students in Singapore studying for 14 hours a day, and how we certainly don not want that to happen. The author conveniently ignores the fact that Finland also scores in the top every single year, and their students have short school days, short semesters, and little or no homework. They still manage to teach their students to read and write. Clearly, it is possible to have a world-class educational system with a reasonable workload for students.

The PISA measures extremely basic skills, without which it doesn’t matter how creative you are. Creativity does not appear from a vacuum. All clever, creative ideas ever have been developed by people who were extremely familiar with what we already know and can do, and based on that figured out what the next step should be. Without previous knowledge of the world, we will just keep inventing the wheel over and over again. It doesn’t matter how smart and creative you are if you are unaware of the world we currently live in. There are plenty of very clever, very motivated, very creative people who spend their time coming up with elementary proofs of Fermat’s last theorem. However, since they lack the formal mathematical training, these proofs are full of elementary mistakes that formal training could teach them to avoid. There are very clever, very motivated aspiring physicists who keep inventing cold fusion. Who knows, maybe if they had spent some time first learning what we have already tried and why that failed, we would be closer to a functioning fusion reactor.

There can be no true creativity without knowledge. Creativity consists of solving problems. As a bare minimum, you need to know what problems are interesting to solve before making great, creative discoveries. To actually find a useful solution, you also need to know exactly what has been tried before, and why that doesn’t work. Without the background knowledge, you will just keep reinventing the wheel. As Newton said: “If I have seen further than others, it is because I stood on the shoulders of giants.” His creative, brilliant insights were based on the knowledge of countless generations before him. As it always has been. As it always will be.

How do you feel about standardised tests and the PISA? Can there be creativity without knowledge?

Top 5 resources for minimalism

During the past year, I’ve been experimenting with minimalism. I know there is some confusion out there about what minimalism really is. That’s because there are two different things people may refer to when they talk about minimalism: a minimalist style and a minimalist lifestyle. The two are not the same thing. There are fashion bloggers with a minimalist style that own 15 white button-downs so that they can pick the one with exactly the right detailing for each outfit. There are people that love colour, antiques, fussy details, and still live a minimalist lifestyle.

A minimalist lifestyle is simply about having only what you need and love. How much this is depends entirely on you. There are minimalists who take it to the extreme and have only 100 possessions, and minimalists whose homes look like yours or mine, just without all the clutter.

For these purposes, clutter is defined as anything you don’t actually need, that simply takes up space and doesn’t make you happy when you look at it. The and is key here. You can have useful things that don’t make you happy when you look at them, but that you still use on a regular basis. Some practical things fall in this category. On the other hand, you have things that serve no purpose than to make you happy, like art, your book collection, and the family photos on the wall. Ideally, most of your possessions should fall in both categories, but nothing should fall in neither.

Clutter is things like the second set of wine glasses that you never use because you don’t like them, the scented candle that you don’t burn because you don’t like the scent, and the fancy clothes steamer that you don’t ever actually use. You may see a theme here: things you don’t use. Things that serve no purpose. In a minimalist lifestyle, these things have no place.

That’s what minimalism is all about: getting rid of things that serve no purpose. And not just physical possessions, but also social commitments, expenses, hobbies, and habits that don’t serve any purpose. If they don’t have any purpose, then you get rid of them. And it’s completely up to you to determine what purposes are worthwhile.

The end result? Less clutter, more time and money to do what really matters to you. Does this sound tempting? Here are the five best resources I’ve found to help with my journey to minimalism.

Into Mind The entire blog is amazing, but I particularly recommend their 30-day minimalism challenge for anyone who wants to get a jump-start on minimalism.

Zen habits This is the blog of minimalism guru Leo Babauta. He writes interesting, thought provoking posts about minimalism on a weekly basis. Some of my favourites are The myth of discipline, Simple daily habits that ignite discipline, and Beating the anxiety of online reading.

The blissful mind Once again, the entire blog is amazing, but in particular Catherine is the one who inspired me to start setting 3 simple goals every month. She also has some great tips for bloggers in this post. And, of course, her blog is a work of art.

Your money or your life by Vicki Robin and Joe Dominguez. This book changed my life. It made me realise that whenever I buy something, I’m trading however much of my life it took me to earn the money, for the item. I’m not trading money for an item, I’m trading parts of my life that I will never get back. I still buy things, but I’m much more aware of what the actual cost is now. I can’r recommend this book enough, even though some parts of it is slightly dated (it was written at a time when you actually earned interest on your savings…).

Lessons from Madame Chic by Jennifer L. Scott. This is a compilation of 20 lessons the (American) author learned from her host family during 6 months in Paris. It’s funny, thought provoking, and teaches you to enjoy all the little things in life. Placing the strawberries just so on the fruit tart you’re cooking. The slice of camembert after dinner. The cold wind on your face when walking to the store. And this is what minimalism is really all about. Not just paring down your life to what you actually consider essential, but also taking the time to enjoy what’s left. Jennifer also has a blog, The Daily Connoisseur.

What blogs / books / podacasts about minimalism do you enjoy?

What’s with all the calculus?

I hate calculus. So, so much. I hate that the first type of mathematics students see at university (the only kind for many of them) is calculus. No one does research in trigonometry. Real analysis is a vibrant research area, but modern real analysis has little to do with the series of tricks we expect students to learn in calculus class. That’s actually what annoys me the most: the calculus curricula are not about understanding calculus, but rather memorizing a series of tricks to then regurgitate them during the 4 (!) midterms and the final.

If I had a say in the development of mathematics courses at university, this is what I’d do:

Scrap calculus. The general math requirement would be fulfilled by a general introduction to mathematics. We would study the unit circle, coordinate geometry, basic number theory, basic combinatorics, the idea of limits, some interesting probability theory, and some graph theory. This course would introduce the breadth of mathematics, and be a proof-based course.

For science majors, there would be a calculus sequence that prepared them for engineering, physics, and applied mathematics. Since only science majors would take this class, it could be taught considerably faster than current calculus classes. The first semester would cover advanced trigonometry, derivatives, and integrals, the second multivariable calculus, and the third differential equations. This could be well integrated with the science courses to make it even more relevant to the students that actually need to take the calculus sequence.

Math majors would not waste two years taking calculus and differential equations before getting to the real math. Instead, they would take introduction to mathematics, and the first course in the calculus sequence. During their second semester, they would be encouraged to take courses like real analysis, linear algebra, abstract algebra, combinatorics, probability theory and number theory in parallel with the second calculus course.

All courses would be proof-based, for both math majors and non-majors. Any class that consists of learning and regurgitating a series of tricks does not belong at university, and I’m a bit ashamed that I’m TAing one right now. We skip all of the beautiful theory underlying calculus, and simply test the students on how well they can regurgitate tricks in a test situation. I read something wonderful on one of my favourite blogs, Math with Bad Drawings, the other day: math tests should be like Turing tests. They should test if there is something intelligent on the other side; something more than what a computer can do. I feel that the same should hold for math classes in general. If they don’t teach you anything that a computer can’t do, the class should not exist. And that’s why I want to scrap calculus as the introductory math class.

Many students sadly don’t have the background or motivation to learn the interesting parts of calculus, so introductory calculus classes focus on just teaching tricks instead. Moreover, the calculus sequence leaves students with a warped idea of what maths and maths research are. It’s not about tricks or calculating values. It’s about understanding how it all fits together, and how to use small pieces that other people found to build bigger things. I wish that the introductory math classes introduced the feeling you get when you managed to get all the pieces to fit together, and prove something for the first time.

What do you think?

A theorem by Erdös

This is a quick application of the probabilistic method to additive number theory that I personally think is really cool. A set is called sum-free if it contains no such that .

Theorem (Erdös, 1965). Every set of nonzero integers contains a sum-free subset of cardinality at least .

Proof. Let be a prime such that and let . Clearly, is a sum-free subset of the group , and .

Let us choose at random an integer , according to a uniform distribution on and define by $d_i \equiv xb_1 \pmod p$, and let $latex 0\leq d_i

1/3$. Therefore, the expected number of elements such that is more than .

Consequently there is an and a subsequence of of cardinality at least such that for all . Since is sum-free, so is . Q.E.D.

Update on my October goals

This far, I’ve been mostly good. I’ve remembered to bring lunch 4 days out of 5 every week. To improve that, I’m modifying that goal to include making the lunch the night before. I tend to not have too much time in the mornings (hello 8 am recitations), so it will be easier to bring my lunch sandwich if all I have to do is grab it from the fridge.

As for not buying anything unnecessary: I’ve bought a couple of things for my Halloween costume and a new eye shadow / eye liner after my old one ran out. I’m mostly being good though. Mostly. I’m new to the whole minimalism thing, and not buying things is a lot harder than it might seem. I’ve managed to stay away from buying books (hello library) and clothes though, and that’s a big win for me. I can’t stop myself from buying books, so I’ve stayed away from Amazon and all the charming bookstores in Hillsboro.

I’ve worked on the probabilistic method 4-5 days every week. I’m not reaching my target of 6 days a week (Saturdays are no math days for me). This is one I’ll have to work much harder on. The book is fascinating, but I’m failing at carving out time for it. From now on, my work in that book will be scheduled every day, so I can’t fail so easily.

The probabilistic method is an extremely clever idea. The basic idea is that if we want to prove that something exists, we don’t need to construct it. We simply construct a probability space, and show that if we randomly select an object from the space, there is a positive probability that we chose our desired object. Since there is a positive probability to chose it, we know that it must exist. I’m currently writing up a post on some examples of the probabilistic method that should go up within a few days.

My 3 goals for October

I don’t do New Year’s promises. They are usually overly ambitious, and realy on the idea that we should only try to improve once every year. Instead, I do monthly goals, and evaluate half-way through. This month, I’m sharing my 3 most important goals for the month.

1 I will work in “The Probabilistic Method” every day. I’m getting really interested in probabilistic methods and extremal graph theory, and this is the book to read in that area.

2 I will bring lunch to work at least 4 days every week. Sigh. I usually plan this out really well, and then get too lazy to prepare lunch in the morning. So I’m going to prepare lunch the night before to eliminate that excuse.

3 I will not buy anything new (except Christmas presents) all month. Minimalism challenge: level 1.

As you can tell: a combination of work-related and personal goals. I find that limiting myself to 3 makes me more likely to succeed than tons of goals. That’s why I try to have one related to work, one related to good habits, and one related to minimalism every month. I’ll let you know how I’m doing in 10 days!

On academic dress

As a somewhat stylish young female scientist, I find myself thinking about the academic dress code. Long gone are the days when academic robes were required to lecture, and the dress code “academic slob” has taken its place. I’ve had lecturers who wear 20 year old t-shirts with holes that show their belly, and lecturers that wear flip flops and khaki shorts every day. Only a single one of my current professors wears slacks and a button-down on a daily basis. And this is most visible in the STEM subjects.

I care about what I wear. There. I said it. There is an attitude in STEM that someone who cares about what they wear is automatically a less competent researcher. However, I was raised to think that wearing nice clothes is a way to show respect to your surroundings and the job your are there to do. Putting on a nice work-appropriate summer dress and a pair of matching ballet flats is a way to show that I care about my job, and that I respect my fellow academics and students. To me, showing up to work looking like you just finished weeding the garden is disrespectful. You don’t show any concern for your coworkers, your students, or your profession dressed in “academic slob”. It doesn’t take longer to put on a dress or a pair of khakis and a button-down shirt (they make wrinkle-resistant ones now, so you don’t even have to iron anything) than it takes to throw on a pair of worn shorts and a holey t-shirt. There is literally no more effort required, but you immediately show that you take yourself, your coworkers, and your profession seriously.

I’m not saying that men should wear a tie and women pencil skirts. That would be excessive. However, would it really be that bad to care just a little that you look like a professional? Tho care how you come across? An argument I’ve heard often why academics should dress down is that it makes us more approachable. That’s bullshit. The only thing I’ve felt and heard from others when faced with yet another lecturer in a tshirt that bares part of a hairy belly is disappointment. It makes students feel like you don’t even care enough about the lecture to put on clothes that fit. If you know your subject, students will respect you for that, but they still feel like you don’t care, so why should they make an effort? Dressing nice is simply a sign of respect for you environment. Nothing more, and nothing less.

Could we all agree to make an attempt? Change the standard from “academic slob” to “academic chic”, and once again take pride in our profession. Put on some clothes that you wouldn’t wear to the gym, or to weed your garden. Some clothes that actually fit you, not clothes that fit the body you had 20 years ago. Showing a sliver of belly is not professional for men or women. It does not make you more approachable. It makes you look less professional. It makes you look like you don’t care about your job, your students or your coworkers.